Chapter 4 & 5 Dynamics: Newton's Laws and Its Application

2 Isaac Newton ( ) “The discovery of the laws of dynamics, or the laws of motion, was a dramatic moment in the history of science... after Newton there was complete understanding. —— R. Feynman “Nature and Nature's laws lay hid in night; God said, Let Newton be! And all was light.” Mathematician and physicist, the greatest scientist of all time.

3 Newton’s laws Work and energy Momentum Angular momentum “If I have been able to see a little farther than other man, it’s because I have stood on the shoulder of giants.” —— —— Isaac Newton

4 Force G m F F G (action capable of accelerating an object)

5 Newton’s First Law of Motion Aristotle ( B.C. ) Force is necessary to keep a body in motion Galileo Galilei ( ) Idealized experiment with no friction No forces → constant speed in a straight line Net force F1F1 F2F2

6 Newton’s first law of motion: Every body continues in its state of rest or of uniform speed in a straight line as long as no net force acts on it. Inertia: tendency to keep one’s state of motion Inertial reference frames Noninertial reference frames How to distinguish? By checking to see if Newton’s first law holds.

7 Mass Mass is a measure of the inertia of a body. More mass ←→ harder to change its velocity ◆ Mass and weight Unit of mass: kilogram (kg)

8 Newton’s Second Law of Motion No forces → constant speed in a straight line velocity will change Newton’s second law of motion: The acceleration of an object is directly proportional to the net force acting on it and is inversely proportional to its mass. The direction of the acceleration is in the direction of the net force acting on the object.

9 Component form in rectangular coordinates Unit of force is Newton (N) 1N=1kg·m/s 2 Limitations: Macro world, low speed motion, inertial frames

10 Newton’s Third Law of Motion Where do forces come from? Newton’s third law of motion: Whenever an object exerts a force on a second object, the second object exerts an equal and opposite force on the first. F1F1 F2F2 Action ←→ reaction ◆ Action force and reaction force are acting on different objects!

Weight and normal force 11 Weight: the magnitude of gravity Direction: down toward the center of Earth Contact force: ∥ Friction ⊥ Normal force g = 9.80m/s 2

Weight loss 12 Example1: Someone reads his weight 600N in an elevator when it accelerates at 0.2g to the ground, what is the real weight? a =0.2g Solution: F N =600N What about this acceleration? a=g

Friction 13 Kinetic friction Static friction Why there is a friction? (P105, Figure 5-1) coefficient of kinetic friction coefficient of static friction and depend on the surfaces: materiel, smoothness, lubrication, …

14 How to solve problems 1. Identify all the bodies 2. Draw a free-body diagram, show all the forces 3. Choose a convenient x-y coordinate system 4. Component equations of Newton’s second law 5. Solve all the equations ▲ Be careful about limitations of the formulas!

Tensions in rope 15 Example2: An object (m=10kg) is hanged on a rope as the figure. What are the tensions in the rope? (Ignoring the mass of rope) Solution: 1) free-body diagram m 60°30° mg F T1 F T2 2) coordinate system x o y 3) component equations:

*Mass of rope & Catenary 16 Consider a rope with uniformly distributed mass, what is the shape of a hanging rope? Catenary Search and read literatures if you are interested

Pulling a box 17 Example3: Someone is pulling a box, where µ=0.75, h=1.5m. If he wants to move with constant velocity and exert a minimum force, L=? Solution: m F µ h L  mg FNFN FµFµ x o y L=h/sin  =2.5m

Broken Atwood’s machine 18 Example4: Two masses connected by a rope and a pulley (Atwood’s machine). The connection part of m 2 is broken and m 2 is moving with constant acceleration a 0 relative to the rope. What are the accelerations of m 1 and m 2 relative to the ground? (Ignoring the mass of rope and pulley) m1m1 m2m2 T m1gm1g m1m1 aoao m2gm2g T m2m2 Solution: a1a1 a2a2 m 1 : m 1 g - T = m 1 a 1 m 2 ： m 2 g - T= m 2 a 2

19 m 1 : m 1 g - T = m 1 a 1 m 2 ： m 2 g - T = m 2 a 2 or ： a 0 = a 1 + a 2 Solve these equations: T m1gm1g m1m1 aoao m2gm2g T m2m2 m1m1 m2m2 a1a1 a2a2

Uniform circular motion 20 Radial acceleration Net force oo r Centripetal/radial force Not some new kind of force It can be tension, weight, friction or the net force of these kind of forces. Determined by a R

Conical pendulum 21 Example5: conical pendulum. a) What is the direction of acceleration? b) Calculate the speed and period. O m L Solution: a) direction of acceleration FTFT G b) horizontal: vertical: r

22 R Example6: A thin circular horizontal hoop of mass m and radius R rotates at frequency f. Determine the tension within the hoop. Rotating hoop Solution: Consider a tiny section of the hoop with an angle d , it has a mass speed Centripetal force dd T T Tension:

Highway curves 23 Unbanked curves Centripetal force: friction Banked curves Centripetal force: Net force of N and G (and friction) N G F Where does centripetal force come from? F v

R   24 Example7: Someone is skating in a bowl shape track with constant , determine the height h. (no friction) h Solution: horizontal: vertical: Ncos  =mg Nsin  =m  2 Rsin  Limitation of model N mg Skating in a bowl

Nonuniform circular motion 25 Tangential acceleration Radial acceleration Components of the net force: a tan aRaR Total acceleration

26 Solution:  R o N mg  So: a tan = g sin  Example8: An object (mass m) slides down along a bowl from position A with no friction, find the radial and tangential acceleration of the object. Radial: Tangential: Conservation of energy What is the normal force N ? A

Velocity-dependent force 27 Question: An object falls from rest, under the action of gravity and the air friction F = -  v, what is the speed at time t, and when t→∞? Solution: Separation of variables Using initial conditions t→∞?

 28 Solution: Example9: An object moves along a semi-circular wall on smooth horizontal plane. Known: v 0 and µ, determine its speed v at time t. v o v0v0. µ N fµfµ Radial force Tangential force Speed: R

*Shooting problem 29 A cannon ball is fired with v 0 at angle , and the air friction F = -  v, how to describe the motion, and what is the range? F G x o y

30 How about F = -  ’ v 2 ?